Shutting down a production line, and it is your responsibility to dispose of the extrusion machine

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1. Your company has a customer who is shutting down a production line, and it is your responsibility to dispose of the extrusion machine. The company could keep it in inventory for a possible future product and estimates that the reservation value is $200,000. Your dealings on the secondhand market lead you to believe that if you commit to a price of $300,000, there is a 0.5 chance you will be able to sell the machine. If you commit to a price of $400,000, there is a 0.2 chance you will be able to sell the machine. If you commit to a price of $500,000, there is a 0.1 chance you will be able to sell the machine. These probabilities are summarized in the following table.
For each posted price, enter the expected value of attempting to sell the machine at that price. (Hint: Be sure to take into account the value of the machine to your company in the event that you are not be able to sell the machine.)
Posted Price Probability of Sale Expected Value
($) ($)
$500,000 0.1 $__?
$400,000 0.2 $__?
$300,000 0.5 $__?
Assume you must commit to one posted price.
In order to maximize the expected profit of the potential sale, which posted price would you commit to in order to maximize the expected value of the potential sale of the machine?
$300,000
$500,000
$400,000

2. Suppose that every driver faces a 5% probability of an automobile accident every year. An accident will, on average, cost each driver $10,000. Suppose there are two types of individuals: those with $80,000.00 in the bank and those with $2,000.00 in the bank. Assume that individuals with $2,000.00 in the bank declare bankruptcy if they get in an accident. In bankruptcy, creditors receive only what individuals have in the bank. Assume that both types of individuals are only slightly risk averse.
In this scenario, the actuarially fair price of full insurance, in which all damages are paid by the insurance company, is $_____?
.
Assume that the price of insurance is set at the actuarially fair price.
At this price, drivers with $80,000.00 in the bank likely will/will not? buy insurance, and those with $2,000.00 in the bank likely will/will not? buy insurance. (Hint: For each type of driver, compare the price of insurance to the expected cost without insurance.)
Suppose a state law has been passed forcing all individuals to purchase insurance at the actuarially fair price.
True or False: The law will affect only the behavior of drivers with $2,000.00 in the bank.
False
True

Sample Solution

Problem 1

Expected Value Calculation

Expected Value = (Probability of Sale * Sale Price) + (Probability of No Sale * Reservation Value)

  • $500,000 posted price:

    • Expected Value = (0.1 * $500,000) + (0.9 * $200,000) = $50,000 + $180,000 = $230,000

  • $400,000 posted price:

    • Expected Value = (0.2 * $400,000) + (0.8 * $200,000) = $80,000 + $160,000 = $240,000

  • $300,000 posted price:

    • Expected Value = (0.5 * $300,000) + (0.5 * $200,000) = $150,000 + $100,000 = $250,000

Optimal Posted Price

To maximize the expected profit, the company should choose the $300,000 posted price, as it yields the highest expected value of $250,000.

Problem 2

Actuarially Fair Price of Insurance

The actuarially fair price of insurance equals the expected cost of an accident:

  • Actuarially Fair Price = Probability of Accident * Average Cost of Accident
  • Actuarially Fair Price = 0.05 * $10,000 = $500

Insurance Purchase Decision

  • Drivers with $80,000:

    • Expected cost without insurance = 0.05 * $10,000 = $500
    • Since the insurance price is equal to the expected cost without insurance, these drivers are indifferent between buying and not buying insurance. However, given their slight risk aversion, they are likely to buy insurance.

  • Drivers with $2,000:

    • Expected cost without insurance = 0.05 * $2,000 = $100 (since they only have $2,000)
    • The insurance price of $500 is significantly higher than the expected cost without insurance. Therefore, these drivers are likely to not buy insurance.

Impact of Mandatory Insurance

  • True or False: The law will affect only the behavior of drivers with $2,000 in the bank.
    • True. Only the drivers with $2,000 in the bank will be affected by the mandatory insurance law, as they would not have chosen to buy insurance voluntarily.

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